We present an extension of the work of D'Amato and Pastawski [Phys. Rev. B 41, 7411 (1990)] on electron transport in a one dimensional conductor modeled by the tight-binding lattice Hamiltonian and in which inelastic scattering is incorporated by connecting each site of the lattice to one dimensional leads. This model incorporates Büttiker's [Phys. Rev. B 32, 1846 (1985); 23, 3020 (1986)] original idea of dephasing probes. Here, we consider finite temperatures and study both electrical and heat transport across a chain with applied chemical potential and temperature gradients. Our approach involves quantum Langevin equations and nonequilibrium Green's functions. In the linear-response limit, we are able to solve the model exactly and obtain expressions for various transport coefficients. Standard linear-response relations are shown to be valid. We also explicitly compute the heat dissipation and show that for wires of length N>>[script-l], where [script-l] is a coherence length scale, dissipation takes place uniformly along the wire. For N<<[script-l], when transport is ballistic, dissipation is mostly at the contacts. In the intermediate range between Ohmic and ballistic transport, we find that the chemical-potential profile is linear in the bulk with sharp jumps at the boundaries. These are explained using a simple model where the left and right moving electrons behave as persistent random walkers.